Non-linear optimal stopping and games with Bermudan strategies

We address an optimal stopping problem, a game problem, and a multiple stopping problem over the set of Bermudan-type strategies ⇥ (which we understand in a more general sense than the stopping strategies for Bermudan options in finance) and with non-linear operators (non-linear evaluations) assessing the rewards, under general assumptions on the non-linear operators ⇢. We provide a characterization of the value family V of the single-agent optimal stopping problem in terms of what we call the p⇥,⇢q-Snell envelope of the pay-o↵ family. We establish a Dynamic Programming Principle. We provide an optimality criterion in terms of a p⇥,⇢q-martingale property of V on a stochastic interval. We investigate the p⇥,⇢q-martingale structure and we show that the ”first time” when the value family coincides with the pay-o↵ family is optimal. The reasoning simplifies in the case where there is a finite number n of pre-described stopping times, where n does not depend on the scenario !. We provide examples of non-linear operators entering our framework. We then explore a non-zero-sum game with two players where each player uses Bermudan strategies to optimize a non-linear assessment functional of the pay-o↵. Using a recursive approach, we show that the game has a Nash equilibrium point. Finally, we consider the non-linear multiple optimal stopping problems with Bermudan strategies. We first study the non-linear double optimal stopping problem by using a reduction approach. We provide a necessary and a sufficient condition for optimal pairs, and a result on existence of optimal pairs. We then generalize the results to the non-linear d-optimal stopping problem. We treat the symmetric case (of additive and multiplicative reward families) as examples.